Math is hard

Hard though you might think math, there is a part of your brain that has no problem doing highly complex differential calculus almost instantaneously. Now before you ask me what I'm on, let me elaborate a bit. Consider a common physics problem: A projectile is launched at a given velocity -- assuming no friction, calculate it's path. Some of us may find that question difficult, some may not. Now let's add in air friction. The question suddenly becomes more complicated. Add in a wind blowing at a given speed in a given direction and the complexity becomes magnified again.

Now, have someone throw a ball at you outside in the wind. Do you have any problems catching it? For most people the answer would be no. True, we may not be given precise information, and we may not be able to say how long the ball will stay in the air, when it reaches it's maximum height, or how fast it's going -- but we can catch it. We can quickly calculate the path it will take and react in order to catch the ball. It may not be the kind of math we normally think of, but it is math nonetheless. Math that functions so quickly and elegantly that we can instantly recieve the pertinent information.

Halfway through writing my response to Andukar, I noticed the appearance of Anark's writeup, which says most of what I was thinking, and more clearly than I could have put it.

I'd have to disagree with Andukar's statement "it is math nonetheless". Mathematics
and physics describe the world, but they aren't actually the world.

Take as an example an equation describing the y-component of projectile motion in two
dimensions:

vy = v0 sin θ + ayt

If someone were to fire a projectile, its motion will, in fact, be precisely predicted by
that equation.1 But is the projectile actually using the equation to
move? Is the universe itself actually running numbers through equations to determine what
it will do next? Clearly not, as this would require an endless stack of places-to-run-the-numbers, which themselves would have to operate on such rules. (cf. Douglas Hofstadter)

The equations describe the events of the universe; they do not specify
the events.

1With the usual caveats about air resistance, relativistic speeds, external
forces, etc., which are not relevant to the discussion.

Consider: when you look at a ball coming towards you in the air, it appears as a point in your two-dimensionalvisual field. Now, if the ball is to the left of the center of your visual field, you move left; if it is to the right of the center you move right; if it is above the center you move back; and if it is below the center you move forward. If you keep the ball in the center of your visual field, it will run right into your face, unless you have a glove in front of it (hopefully you will!).

If you don't believe me, try catching fly balls with only one eye open sometime. It's hard, because with no depth perception you can't figure out the ball's velocity or when it's going to hit you. There has been quite a bit of research which indicates that this is, in fact, the actual algorithm which baseball players use to catch fly balls.1

So can we generalize anything from this? Not conclusively, but I would argue that this and other similar phenomena indicate that we do not in fact do nearly as much computation as we sometimes appears to. (Walking is another example that comes to mind; nowhere in our brain is the trajectory and landing point of our foot calculated when we take a step; we simply swing our leg and let the physics of gravity and pendulums take care of the details.) On the other hand, some behaviors, such as language and social behavior, are different; we do do a lot of complicated unconscious processing for these things. The difference is that evolution has kindly put into our brains a dedicated "language computer", but not a dedicated "mathematics computer". The ability we as humans have in mathematics (which is not inconsiderable) is due to the more general cognitive capacities (catgorization, generalization, etc.) that we have. So, I would say, math really is hard. On the other hand, I think it's also a lot of fun, and I think it would get boring fast if it was always as easy as breathing.

1. I regret that I can't give any references to this research. A web search provides some discussion of this phenomenon, but no links to the original research. If anyone could provide me with some references, please /msg me.

Barbie dolls have long been thought of as a grotesquely stereotypical "perfect"
woman in some circles--those disgusted with Barbie's perfect proportions
note that an anatomically correct life-size Barbie would be 7 feet tall
with a 40-inch bust, 22-inch waist and 36-inch hips...and her legs would
be 5 feet long. But, it took an incident in the late 1980's to bring
mainstream attention to this issue for the first time:

Mattel released a Teen Talk Barbie which could say over 200 different
sentences. Among these were "Math is hard!" and "Let's go shopping!"
Activist groups throughout the country were outraged that Mattel would
program Barbie with such overtly sexist sayings, and the doll became
the butt of countless jokes in the media and on the late night talk
show circuit.

Mattel quickly pulled the dolls from the shelves and reprogrammed Barbie
without the offensive sayings, but the damage had already been done--many
more people than ever before began to see that Barbie might just be everything
that its opponents had been claiming for many years--materialistic, impossibly
proportioned, and subservient.